On a class of resolvable Lie algebras (Q1373015)

Let \(L\) be a real finite-dimensional Lie algebra, \(G\) the corresponding simply connected Lie group and \(\text{Ad}:G\to \Aut(L)\) be its adjoint representation in \(L\). Let \(G\) act on the dual space \(L^*\) of \(L\) by \(\text{Ad}^*:G\to \Aut(L^*)\) by \(\langle \text{Ad}_g^*(f),x\rangle= \langle f,\text{Ad}_g- 1(x)\rangle\) for \(f\in L^*\) and \(x\in L\), \(g\in G\). Call this the \(K\)-representation of the Lie group \(G\) and its orbits in \(L^*\) the \(K\)-orbits. The Lie algebra \(L\) is said to satisfy the MD property if the dimension of every \(K\)-orbit in \(L^*\) is 0 or equal to \(\dim L\). The author investigates the properties of such Lie algebras and classifies all Lie algebras of the class MD with dimension \(\leq 5\).